ABSTRACT
This book completely resolves the problems, which are induced by the lacunae of the theory, for all three classes of the systems. To be specific:
It discovers, defines and effectively exploits the state variables, i.e., the state vector, of both IO and IIO systems. They have the full physical sense. This permitted to extend the (Lyapunov and Bounded-Input, or BI) stability concepts and properties directly to these classes of the systems. This enabled their direct stability study rather than to study them formally mathematically, without any physical sense, in their formally mathematically (without a physical sense) transformed form of the ISO systems.
It unifies the study and applications of all three classes of the systems. This is due to the following contributions that hold for all three classes of the systems.
It discovers a complex domain fundamental dynamical characteristic of the systems, which is their full transfer function matrix F(s), and which in the domain of the complex variable s shows how the system in the course of time transmits and transfers the simultaneous influence of the system input vector and of all initial conditions on the system output (or on the system state).
It presents the definition of F(s) and completely resolves the problem of its determination so that it has the same properties as the well-known system transfer function matrix G(s): its compact matrix form, 364its independence of the external actions and of initial conditions acting on the system, i.e., its dependence only on the system order, structure and parameters.
It shows the physical meaning of F(s) and its link with the complete time response of the system.
It establishes the full block diagram technique based on the use of F(s), which incorporates the Laplace transform of the input vector and the vector of all initial conditions, and which generalizes the well-known block diagram technique.
It exactly and completely resolves the problem of the pole-zero cancellation.
It introduces the concept of the system full matrix P(s) in the complex domain and establishes its link with the system full transfer function matrix F(s).
It defines the system equivalence under nonzero initial conditions and proves the related conditions.
It establishes the direct relationship between the system full transfer function matrix F(s) and the Lyapunov stability concept, definitions and conditions. This shows that the Lyapunov stability test via the system transfer function matrix G(s) can end with crucially wrong result. This refines inherently the complex domain criteria for the Lyapunov stability properties.
It extends, broadens and generalizes the BI stability concept by introducing new BI stability properties that incorporate nonzero initial conditions, presents their exact definitions and proves the conditions for their validity in terms of the system full transfer function matrix F(s) and its submatrices.
